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has been cited by the following article:

• TITLE: Existence of Solutions for Shallow Water Equatorial Waves

  AUTHORS: Ranis Nail Ibragimov, Svetlin Georgiev, Deliang Bao, Joy Williams

  KEYWORDS: Shallow Water, Equatorial Waves, Existence of Solutions, Integral Presentation

  JOURNAL NAME: Journal of Applied Mathematics and Physics, Vol.14 No.2, February 10, 2026

  ABSTRACT: The study focuses on the modeling of nonlinear shallow water equatorial free boundary waves. It has been demonstrated that there exists at least one solution, at least two nonnegative solutions, and at least three nonnegative solutions within the time scales considered. To achieve this, the original nonlinear model is transformed into a linear system using the hodograph method. Subsequently, the mapped system is represented through a new integral formulation that involves two operators. These operators are designed to ensure that any fixed point of their sum corresponds to a solution of the problem at hand.